Finite element θ-schemes for the acoustic wave equation

Samir Karaa*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1 Δ+ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and Δt the time step.

Original languageEnglish
Pages (from-to)181-203
Number of pages23
JournalAdvances in Applied Mathematics and Mechanics
Issue number2
Publication statusPublished - 2011


  • Discontinuous galerkin methods
  • Energy method
  • Finite element methods
  • Implicit methods
  • Optimal error estimates
  • Stability condition
  • Wave equation

ASJC Scopus subject areas

  • Mechanical Engineering
  • Applied Mathematics


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